Difference between revisions of "Banach limit"
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2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009029.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009030.png" />), where, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009031.png" /> is the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009032.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009033.png" /> itself is called left (respectively, right) amenable if there exists a left- (respectively, right-) invariant mean in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009034.png" />. The existence of Banach limits above is a special case of an invariant mean, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009035.png" /> equals the semi-group of natural numbers. Banach also proved that the real numbers are amenable (left and right). M.M. Day has proved that every Abelian semi-group is left and right amenable. On the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009036.png" />, the free group on two generators, is not amenable. | 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009029.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009030.png" />), where, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009031.png" /> is the adjoint of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009032.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009033.png" /> itself is called left (respectively, right) amenable if there exists a left- (respectively, right-) invariant mean in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009034.png" />. The existence of Banach limits above is a special case of an invariant mean, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009035.png" /> equals the semi-group of natural numbers. Banach also proved that the real numbers are amenable (left and right). M.M. Day has proved that every Abelian semi-group is left and right amenable. On the other hand, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009036.png" />, the free group on two generators, is not amenable. | ||
− | Another approach to amenability is the measure-theoretic point of view. In fact, the prehistory of amenability starts with the following question by H. Lebesgue in the classic "Leçons sur L'Intégration et la Recherche des Fonctions Primitives" ([[#References|[a5]]], pp. 114–115): Can countable additivity of the [[Lebesgue measure|Lebesgue measure]] be replaced by finite additivity? Banach answered the question in the negative, constructing a finitely additive measure on all subsets of the real numbers, invariant under translation, again using the Hahn–Banach theorem. More generally, if a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009037.png" /> is acting on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009038.png" />, a finitely additive [[Probability measure|probability measure]] on the collection of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009039.png" />, invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009040.png" />, is sometimes also called an invariant mean. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009041.png" /> is the isometry group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009042.png" />, one can ask for a finitely additive measure invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009043.png" />. Such a measure does exist for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009044.png" />, but not for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009045.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009046.png" /> this leads to so-called paradoxical decompositions or the Banach–Tarski paradox (see [[Tarski problem|Tarski problem]]; for a survey, see [[#References|[a8]]]). For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009047.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009048.png" /> contains the non-amenable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009049.png" /> as a subgroup. It has been proved that the Banach–Tarski paradox is effectively (i.e., in ZF set theory) implied by the Hahn–Banach theorem (see [[#References|[a7]]]). | + | Another approach to amenability is the measure-theoretic point of view. In fact, the prehistory of amenability starts with the following question by H. Lebesgue in the classic "Leçons sur L'Intégration et la Recherche des Fonctions Primitives" ([[#References|[a5]]], pp. 114–115): Can countable additivity of the [[Lebesgue measure|Lebesgue measure]] be replaced by finite additivity? Banach answered the question in the negative, constructing a finitely additive measure on all subsets of the real numbers, invariant under translation, again using the Hahn–Banach theorem. More generally, if a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009037.png" /> is acting on a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009038.png" />, a finitely additive [[Probability measure|probability measure]] on the collection of all subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009039.png" />, invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009040.png" />, is sometimes also called an invariant mean. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009041.png" /> is the isometry group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009042.png" />, one can ask for a finitely additive measure invariant under <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009043.png" />. Such a measure does exist for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009044.png" />, but not for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009045.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009046.png" /> this leads to so-called paradoxical decompositions or the [[Banach–Tarski paradox]] (see [[Tarski problem|Tarski problem]]; for a survey, see [[#References|[a8]]]). For all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009047.png" />, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009048.png" /> contains the non-amenable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110090/b11009049.png" /> as a subgroup. It has been proved that the Banach–Tarski paradox is effectively (i.e., in ZF set theory) implied by the Hahn–Banach theorem (see [[#References|[a7]]]). |
For a survey of results of the role of amenability, see [[#References|[a6]]] and for a survey of the Hahn–Banach theorem, see [[#References|[a2]]]. For the early history of Banach limits and invariant means, including many important results, see [[#References|[a3]]] and [[#References|[a4]]]. | For a survey of results of the role of amenability, see [[#References|[a6]]] and for a survey of the Hahn–Banach theorem, see [[#References|[a2]]]. For the early history of Banach limits and invariant means, including many important results, see [[#References|[a3]]] and [[#References|[a4]]]. |
Latest revision as of 06:44, 9 October 2016
Banach limits originated in [a1], Chapt. II, Sect. 3. Denoting the positive integers by , the set
is the real vector space of all bounded sequences of real numbers. For any element
, one defines
by
for all
. S. Banach showed that there exists an element in the dual
, called
, such that
1) for all
;
2) for all non-negative sequences
;
3) for all
;
4) for all convergent sequences
. Banach proved the existence of this generalized limit by using the Hahn–Banach theorem. Today (1996), Banach limits are studied via the notion of amenability.
For a semi-group one defines
to be the real vector space of all real bounded functions on
. For an element
one denotes the left (respectively, right) shift by
(respectively,
). Thus,
for all
and
for all
. An element
is called a left- (respectively right-) invariant mean if
1) ;
2) (respectively,
), where, e.g.,
is the adjoint of
.
itself is called left (respectively, right) amenable if there exists a left- (respectively, right-) invariant mean in
. The existence of Banach limits above is a special case of an invariant mean, where
equals the semi-group of natural numbers. Banach also proved that the real numbers are amenable (left and right). M.M. Day has proved that every Abelian semi-group is left and right amenable. On the other hand,
, the free group on two generators, is not amenable.
Another approach to amenability is the measure-theoretic point of view. In fact, the prehistory of amenability starts with the following question by H. Lebesgue in the classic "Leçons sur L'Intégration et la Recherche des Fonctions Primitives" ([a5], pp. 114–115): Can countable additivity of the Lebesgue measure be replaced by finite additivity? Banach answered the question in the negative, constructing a finitely additive measure on all subsets of the real numbers, invariant under translation, again using the Hahn–Banach theorem. More generally, if a group is acting on a set
, a finitely additive probability measure on the collection of all subsets of
, invariant under
, is sometimes also called an invariant mean. If
is the isometry group of
, one can ask for a finitely additive measure invariant under
. Such a measure does exist for
, but not for
. For
this leads to so-called paradoxical decompositions or the Banach–Tarski paradox (see Tarski problem; for a survey, see [a8]). For all
, the group
contains the non-amenable
as a subgroup. It has been proved that the Banach–Tarski paradox is effectively (i.e., in ZF set theory) implied by the Hahn–Banach theorem (see [a7]).
For a survey of results of the role of amenability, see [a6] and for a survey of the Hahn–Banach theorem, see [a2]. For the early history of Banach limits and invariant means, including many important results, see [a3] and [a4].
References
[a1] | S. Banach, "Théorie des opérations linéaires" , PWN (1932) |
[a2] | G. Buskes, "The Hahn–Banach theorem surveyed" Dissertationes Mathematicae , CCCXXVII (1993) |
[a3] | M.M. Day, "Normed linear spaces" , Ergebnisse der Mathematik und ihrer Grenzgebiete , 21 , Springer (1973) |
[a4] | Greenleaf, F.P, "Invariant means on topological groups and their applications" , v. Nostrand (1969) |
[a5] | H. Lebesgue, "Oeuvres Scientifiques" , L'Enseign. Math. , II , Inst. Math. Univ. Genæve (1972) |
[a6] | A.L.T. Paterson, "Amenibility" , Mathematical Surveys and Monographs , 29 , Amer. Math. Soc. (1988) |
[a7] | J. Pawlikowski, "The Hahn–Banach theorem implies the Banach–Tarski paradox" Fundam. Math. , 138 (1991) pp. 20–22 |
[a8] | S. Wagon, "The Banach–Tarski paradox" , Cambridge Univ. Press (1986) |
Banach limit. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Banach_limit&oldid=13656